| # Quantum radiation from a strongly driven two-level system |
|
|
| ## Background |
|
|
| A strongly driven two-level emitter can radiate while ending in either of its |
| bare states. The same radiation may also be discussed in terms of the part |
| that is phase coherent across an ensemble of identical emitters and the part |
| associated with quantum fluctuations. These are different physical |
| resolutions of one emitted state. Their relation under strong driving is the |
| object of this task. |
|
|
| ## Research problem |
|
|
| Develop a quantum-optical calculation that resolves the radiation from the |
| driven two-level system both by the emitter's final state and by radiation |
| coherence. Determine how the two descriptions are related and whether, and if |
| so how, their relationship can be controlled by changing the driving strength. |
|
|
| The derivation, numerical construction, choice of scientific evidence and |
| organization of the argument are yours. Establish the conclusion from your |
| calculation rather than assuming a fixed correspondence between the two |
| descriptions. |
|
|
| ## Physical inputs |
|
|
| The emitter has bare states $|g\rangle$ and $|e\rangle$, energy separation |
| $\hbar\omega_a$, and only the off-diagonal electric-dipole matrix element |
| $\Omega=\langle e|\hat d|g\rangle$ (with its conjugate). It starts in |
| $|g\rangle$. The quantized radiation modes start in vacuum. |
| |
| Treat the classical drive nonperturbatively. Its electric field is |
| |
| $$ |
| E_L(t)=E_0\sin^2\!\left(\frac{\omega_0t}{2N_c}\right) |
| \sin(\omega_0t), |
| \qquad 0\le t\le \frac{2\pi N_c}{\omega_0}, |
| $$ |
|
|
| and is zero outside this interval. Use the same sine-squared envelope as the |
| turn-on and turn-off factor in the coupling to the quantized radiation modes. |
| Treat that coupling to first order; reabsorption and higher orders in the |
| emitted field are outside the task. |
|
|
| Use |
|
|
| $$ |
| \omega_0=\frac{\omega_a}{7},\qquad N_c=20, |
| $$ |
| |
| and the following cases: |
| |
| | Case | $E_0|\Omega|/\omega_0$ | |
| |---|---:| |
| | A | 11.23 | |
| | B | 13.21 | |
| | C | 13.87 | |
| |
| You may set $\hbar=\omega_a=|\Omega|=1$ or use another consistent unit |
| system. State all phase, Fourier and basis conventions. |
|
|
| The graded spectral object is the squared modulus of the first-order emission |
| amplitude before any optional frequency-dependent radiative, |
| density-of-modes or detector prefactor. Declare any constant normalization |
| used in a comparison; absolute spectral magnitude is not a grading target. |
|
|
| ## Method boundary |
|
|
| Use standard two-level dynamics and first-order quantum-optical emission |
| theory. You must derive the task-specific quantities used to resolve the two |
| descriptions and their relationship. No particular propagator, quadrature, |
| grid, transform implementation, plotting layout or validation procedure is |
| prescribed. |
|
|
| You are responsible for checking that the computed evidence is sufficiently |
| reliable and reproducible for the claims made. A state or spectral comparison |
| must be backed by reconstructible numerical data from the corresponding case. |
|
|
| ## Submission |
|
|
| Submit one self-contained scientific report and an offline reproduction |
| package containing the derivation, executable code, numerical data underlying |
| the reported emitter-state and radiation-coherence evidence, references |
| actually used, and concise run and environment information. The choice and |
| number of figures are yours. No fixed file tree, filename scheme, exploratory |
| history or examiner-selected validation sequence is required. |
|
|
| Include a provenance note listing every offline source or software document actually consulted and the method basis for items supplied from general knowledge. |
|
|
| The active examination resources and limits are defined in the companion |
| `resource_limits.json`. All retained production calculations must run |
| within that fixed allocation and active wall time. More expensive modelling |
| does not receive additional credit. |
|
|
| ## Reference and integrity policy |
|
|
| This examination is conducted without network access or live literature |
| retrieval. You may rely on your existing general scientific knowledge and, if any are |
| explicitly supplied with the examination, on those offline materials or |
| software documentation. Record every such source actually consulted. For formulas, |
| methods, constants, or conventions used from general knowledge rather than a |
| consulted source, state enough of the method basis, definition, value, units, |
| and convention to make the calculation auditable; do not invent a citation. |
| You are not required to identify or prove a publication date or to retrieve |
| literature during the examination. |
|
|
| The hidden target article and its identifying metadata, figures or captions, |
| supplementary material, author code or data, target-specific reproduction |
| code, solution or scoring material, online solutions, and worked |
| reproductions of this case must not be used. All submitted scientific figures |
| and numerical data must be generated by your own implementation of the stated |
| problem. |
|
|
| See [submission requirements](../../submission_requirements.md) for the common submission entry points. |
|
|
